Balanced Splitting Methods Infinite Matrices

Date: 
Wednesday, January 4, 2012 - 4:39pm

UCSB COMPUTER SCIENCE DEPARTMENT/CSE IGERT PRESENTS:

Monday, January 30, 2012
3:00 – 4:00 PM
Computer Science Conference Room, Harold Frank Hall Rm. 1132

HOST: Linda Petzold

SPEAKER: Gilbert Strang
MIT

Title: Balanced Splitting Methods / Infinite Matrices

Abstract:

Most of this talk is about a problem that constantly arises in
scientific computing.
The last part is about an algebra problem — for finite and infinite
matrices.

Differential equations often have diffusion and advection and reaction
terms. Those are
treated differently, and separately when possible. Diffusion might be
implicit and advection
explicit. Reaction is highly nonlinear, but local. If we separate the
transport terms T(u)
from the reaction terms R(u), we may “split” each time step into
separate integrations.
Since they don’t commute, the overall step has only first-order
accuracy. But second order
is achieved by a half-step based on T, a full step based on R, and a
half-step based on T.

Problem: An error can appear in the steady state. The solution to T(u)+
R(u) = 0 may not
solve T(u) = 0 and R(u) = 0 separately. Solution: Adjust to T(u) + c_n
and R(u) – c_n by a
balancing vector c_n at each step. If we choose c_n = (R(u_n) -
T(u_n))/2 then each part can
converge to the correct steady state. But stability becomes harder to
ensure — and Ray Speth
has created a “rebalanced splitting” that is equally accurate and much
more stable.

For two problems on banded doubly infinite matrices there is progress to
report:

1. Is the triangular factorization A = LPU still possible ? Notice the
position of P !!

2. Which is the ‘main diagonal’ of that infinite matrix ?

Bio:

Gilbert Strang was an undergraduate at MIT and a Rhodes Scholar at
Balliol College, Oxford. His Ph.D. was from UCLA and since then he has
taught at MIT. He is a Fellow of the American Academy of Arts and
Sciences and the National Academy of Sciences.. Professor Strang has
published eight textbooks including Introduction to Linear Algebra and
Computational Science and Engineering.

He was the President of SIAM during 1999 and 2000, and Chair of the
Joint Policy Board for Mathematics. He received the von Neumann Medal
for computational mechanics, and the Henrici Prize for applied analysis.
The first Su Buchin Prize and the Haimo Prize from the MAA were awarded
for his contributions to teaching around the world.
His video lectures on linear algebra and CSE are on ocw.mit.edu